David A. Herron (University of Cincinnati)
Quasiconformal deformations and volume growth
Abstract.
Bonk-Heinonen-Koskela have produced a uniformization theory which, roughly
speaking,
provides a one-to-one correspondence between (certain equivalence classes
of)
uniform
spaces and proper geodesic Gromov hyperbolic metric spaces. However,
their
uniformization
produces an associated measure which, in general, has exponential volume
growth. We
determine when a metric measure space admits a uniformizing conformal
density
which has
Ahlfors regular volume growth. Under certain mild hypotheses, this occurs
precisely
when the conformal Assouad dimension of the Gromov boundary is small
enough.
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